Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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REPRODUCING KERNEL METHOD FOR SOLVING TENTH-ORDER BOUNDARY VALUE PROBLEMS

  • Geng, Fazhan (Department of Mathematics, Changshu Institute of Technology) ;
  • Cui, Minggen (Department of Mathematics, Harbin Institute of Technology)
  • Received : 2009.08.19
  • Accepted : 2009.09.27
  • Published : 2010.05.30
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Abstract

In this paper, the tenth-order linear boundary value problems are solved using reproducing kernel method. The algorithm developed approximates the solutions, and their higher-order derivatives, of differential equations and it avoids the complexity provided by other numerical approaches. First a new reproducing kernel space is constructed to solve this class of tenth-order linear boundary value problems; then the approximate solutions of such problems are given in the form of series using the present method. Three examples compared with those considered by Siddiqi, Twizell and Akram [S.S. Siddiqi, E.H. Twizell, Spline solutions of linear tenth order boundary value problems, Int. J. Comput. Math. 68 (1998) 345-362; S.S.Siddiqi, G.Akram, Solutions of tenth-order boundary value problems using eleventh degree spline, Applied Mathematics and Computation 185 (1)(2007) 115-127] show that the method developed in this paper is more efficient.

Keywords

References

  1. M.G. Cui and Y.Z. Lin, Nonlinear Numerical Analysis in Reproducing Kernel Space, Nova Science Pub Inc, 2009.
  2. A. Daniel, Reproducing Kernel Spaces and Applications, Springer, 2003.
  3. A. Berlinet and C. Thomas-Agnan, Reproducing Kernel Hilbert Space in Probability and Statistics, Kluwer Academic Publishers, 2004.
  4. F.Z. Geng and M.G. Cui, Solving singular nonlinear two-point boundary value problems in the reproducing kernel space, Journal of the Korean Mathematical Society 45 (2008), 77-87.
  5. M.G. Cui and F.Z. Geng, Solving singular two-point boundary value problem in reproducing kernel space, Journal of Computational and Applied Mathematics, 205(2007), 6-15. https://doi.org/10.1016/j.cam.2006.04.037
  6. F.Z. Geng and M.G. Cui, Solving a nonlinear system of second order boundary value problems, Journal of Mathematical Analysis and Applications, 327(2007), 1167-1181. https://doi.org/10.1016/j.jmaa.2006.05.011
  7. F.Z. Geng and M.G. Cui, Solving singular nonlinear second-order periodic boundary value problems in the reproducing kernel space, Appl.Math.Comput. 192 (2007), 389-398. https://doi.org/10.1016/j.amc.2007.03.016
  8. M.G. Cui and F.Z. Geng, A computational method for solving one-dimensional variable coefficient Burgers equation, Appl.Math.Comput. 188 (2007), 1389-1401. https://doi.org/10.1016/j.amc.2006.11.005
  9. S.S.Siddiqi and E.H.Twizell, Spline solutions of linear tenth-order boundary-value problems, International Journal of Computer Mathematics 68(1998), 345-362. https://doi.org/10.1080/00207169808804701
  10. A.M. Wazwaz, The Modified Adomian Decomposition Method for Solving Linear and Nonlinear Boundary Value Problems of Tenth-order and Twelfth-order, International Journal of Nonlinear Sciences and Numerical Simulation 1(2000), 17-24.
  11. S.S.Siddiqi and G.Akram, Solution of 10th-order boundary value problems using nonpolynomial spline technique, Applied Mathematics and Computation 190(2007), 641-651. https://doi.org/10.1016/j.amc.2007.01.075
  12. S.S.Siddiqi and G.Akram, Solutions of tenth-order boundary value problems using eleventh degree spline, Applied Mathematics and Computation 185(2007), 115-127. https://doi.org/10.1016/j.amc.2006.07.013
  13. E.H.Twizell, A.Boutayeb and K.Djidjeli, Numerical methods for eighth-, tenth-and twelfth order eigenvalue problems arising in thermal instablity, Advances in Computational Mathematics, 2(1994), 407-436. https://doi.org/10.1007/BF02521607
  14. R.P.Agarwal, Boundary Value Problems for High Ordinary Differential Equations, World Scientific, Singapore,1986.
  15. K.Djidjeli, E.H.Twizell and A.Boutayeb, Numerical methods for special nonlinear boundary-value problems of order 2m, Journal of Computational and Applied Mathematics, 47(1993), 35-45. https://doi.org/10.1016/0377-0427(93)90088-S
  16. S.S.Siddiqi and E.H.Twizell, Spline solutions of linear sixth-order boundary-value problems, International Journal of Computer Mathematics, 60(1996), 295-304. https://doi.org/10.1080/00207169608804493
  17. C.L. Li and M.G. Cui, The exact solution for solving a class nonlinear operator equations in the reproducing kernel space, Appl.Math.Compu, 143(2003), 393-399. https://doi.org/10.1016/S0096-3003(02)00370-3